Review of vector spaces

Definition A.1.1A field F is a commutative ring for which 1 ≠ 0 and each nonzero element a ∈ F has an inverse a-1 ∈ F with a · a-1 = 1.

Let F be any field, and let a, b be nonzero elements of F. Then a has an inverse, so if ab = 0 then a-1(ab) = a-1(0) = 0, which implies that b = 0, a contradiction. We conclude that in any field the product of two nonzero elements must be nonzero.

Definition A.1.2Let F be a field. The smallest positive integer n such that n · 1 = 0 is called the characteristic of F, denoted by char(F). If no such positive integer exists, then F is said to have characteristic zero.

If char (F) = n, then it follows from the distributive law that n · a = (n·1)·a = 0 · a = 0, and so adding any element to itself n times yields 0.

Proposition A.1.3The characteristic of a field is either 0 or p, for some prime number p.

Proof. If a field F has characteristic n, and n is composite, say n = mk, then (m · 1)(k · 1) = n · 1 = 0. This is a contradiction, since in any field the product of two nonzero elements is nonzero. ▪